Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

Abstract

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+σ}),\quad &x\in Ω\cr 0=Δv-μv+νu,\quad &x\in Ω\quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary $\partial \Omega$, $\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $\mu ,\nu$ are positive constants, and $\sigma \ge 0$. When $\sigma >0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar{\Omega })$, (0.1) has a unique globally defined classical solution $(u_{\sigma }(x,t;u_0),v_{\sigma }(x,t;u_0))$ with $u_{\sigma }(x,0;u_0)=u_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{μχ^2}{4}, &\text{if $0<\chi \le 2,$}\\ μ(χ-1), &\text{if $\chi >2$.}\\ \end{cases} \end{equation*} When $\sigma =0$, we show that the classical solution $(u(x,t;u_0,0),v(x,t;u_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u_0(x)$ are not small.